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Math Help - Subspaces Projection and R^n

  1. #1
    Member zangestu888's Avatar
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    Subspaces Projection and R^n

    Two questions am not understanding?
    1) It says;

    Are the foloowing subspaces?Support your answer with detials.
    U1={X belong to R^n:AX=3X}

    and

    2)

    We fix a non-zero vector d belonging to R^3, and define U3={X beloninig to R^3: proj(d)(X)=0}, where projd(x) is the projection of X onto d
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  2. #2
    tah
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    Hi, what is A in 1). Is it a matrix, linear map, ..? If yes, the answer is positive and you need only to show that for any X,Y\in U_1\ and\ c\in\mathbb{R},\ then\ X+Y\in U_1\ and\ cX\in U_1
    you can use the same argument for 2)
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  3. #3
    Member zangestu888's Avatar
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    It is a matrix...am not following ur logic? please expand on it if you may i have more trouble with the projection
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  4. #4
    tah
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    To show that S is a subspace of \mathbb{R}^n, it suffice to show it's stable by addition and multiplication by a scalar (real number). So you need to prove that U_1 has these properties using the fact that A(X+Y) = AX+AY and A(cX) = cAX i.e. A is linear. Also multiplication by 3 is linear.

    for 2), proj(d) is also linear.
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  5. #5
    Member zangestu888's Avatar
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    okay i understoood the first part as it is a subspace so itrs true, am still not sure how to show the proj one? i was never to well with projectin?please help
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  6. #6
    tah
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    Projecting of X on a vector d is equivalent to taking the scalar product X.d and form the vector (X.d)d/(d^2) i.e. proj(d)(X) = (X.d)d/(d^2) (d^2 is the square of the norm of d). You should know that scalar product is linear i.e. (X+Y).d = X.d+Y.d and (cX).d = c(X.d)
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